Ömer Kişi

We study statistical and ideal convergence of double sequences in 2-normed spaces under non-negative RH-regular matrix families. For a family of averages over adjacent row pairs, we give a necessary and sufficient condition for ideal convergence of the transforms to recover convergence of the original sequence. The condition involves the ideal and the lower level sets of the coefficient that separates the two entries of each pair. It applies to all double sequences. We characterize separately when a finite quantitative bound exists and determine its optimal constant. An explicit two-member family recovers statistical convergence although neither member does so alone and no finite quantitative bound is available. The same family fails to recover Pringsheim convergence. We also prove a general diagonal Tauberian estimate for bounded sequences, separate statistical convergence of transforms from exceptional-set matrix mass, and establish density-one, Cauchy and filter-set results under explicit hypotheses. Double Cesàro examples illustrate the distinctions between the convergence notions.

Convergence of moving closed sets is usually studied relative to a family of test regions fixed in advance. This may conceal the regions on which a given sequence actually stabilizes. We introduce a sequence-dependent construction in a metric space. For each test set, we take the infimum over all tails of the supremum of the symmetric localized excess between pairs of tail members. The test sets for which this quantity is zero form the stabilization family. We prove that this family is hereditary and closed under finite unions, and hence is a bornology whenever it covers the ambient space. Under neighborhood stabilization and a compactness condition preventing local traces from escaping, we obtain either eventual disappearance from a test region or convergence to a nonempty local closed limit. A nested countable cofinal family then yields compatible local limits and a global cluster set. When this set is nonempty, bornological convergence follows. We further prove that every preassigned bornology supporting neighborhood convergence is contained in the generated neighborhood-stabilization family. Counterexamples separate the construction from global Hausdorff convergence and pointwise convergence of distance functions, and establish that compactness is indispensable. The novelty is that the testing bornology is derived from the tail behavior of the sequence itself. This construction may be useful for moving feasible sets and set-valued approximations that stabilize locally but not globally.

This paper presents a new sequence space of functions, denoted as [ ], which is defined via [ ]-relative uniform convergence and the Jackson [ ]-difference operator. The space combines aspects of [ ]-convergence, relative uniform convergence, and quantum deformation, shedding light on the interaction between [ ]-calculus and [ ]-averaging. We establish that [ ] is a Banach space and satisfies the properties of both [ ]-spaces and [ ]-spaces when equipped with a [ ]-deformed norm. Further, we explore the geometric properties of this space, including strict convexity, uniform convexity, and the Opial-type condition, all of which depend on the parameters [ ] and [ ]. A [ ]-dependent modulus of convexity is formulated, and we establish inclusion relations with classical spaces, such as [ ] spaces and [ ] difference spaces, including the limiting case as [ ]. These results enrich the theory of sequence spaces and present a unified framework that bridges [ ]-convergence, relative uniform convergence, and quantum calculus.

Multiplicative metric spaces provide a natural framework for analyzing problems involving exponential growth or ratio-based phenomena unlike the classical additive settings. In this paper, we introduce the concept of multiplicative [ ]-statistical convergence by using generalized de la Vallée-Poussin means to accommodate sequences with infrequent yet substantial deviations. We utilize the logarithmic relationship between multiplicative and classical metrics to bridge the gap between these two frameworks which facilitates the transfer of analytical properties without relying solely on isomorphism. We characterize multiplicative [ ]-statistically Cauchy sequences and investigate their connection to boundedness and completeness within this topology. Furthermore, we provide a detailed analysis of statistical limit and cluster points to demonstrate that the set of cluster points forms a closed and compact set under specific conditions. These results extend classical summability theory and offer new methodologies for analyzing systems governed by multiplicative calculus.

This paper proposes the concept of double quasi-statistical convergence within the framework of credibility theory. Quasi-statistical convergence, as a generalization of statistical convergence, plays a key role in capturing more generalized and intricate patterns of convergence in fuzzy variables. Extending earlier studies, we develop this notion for double sequences and examine its connections with other convergence types in credibility theory, providing new insights into two-dimensional fuzzy systems

We study strong lacunary harmonic summability of fractional difference sequences in q -rung orthopair fuzzy normed spaces. The fractional difference is defined on its convergence domain, with an explicit sufficient condition and a truncation estimate for bounded Banach-space-valued sequences. In the canonical norm model, the summability condition is equivalent, on bounded transformed sequences, to strong summability of the norms. We prove linearity, uniqueness and subsequence results for proper admissible ideals, and establish a transfer theorem for distinct lacunary partitions whose harmonic symmetric difference is negligible. In complete spaces, a Cauchy criterion is obtained under bounded normalized row mass; this mass bound holds automatically when β = 1. Counterexamples explain the failure of a fixed-centre zero-limit condition and the obstruction caused by unbounded row mass. The general summability results concern the transformed sequence, whereas the domain and truncation estimates use the fractional coefficients explicitly.

This paper will extend some notions, such as convergence, to b-metric spaces with octonion valued b-metric spaces constructed by Qiu et al. Octonion-valued metric spaces are based on modifying the triangle inequality of a semi-metric space by multiplying one side of the inequality by a scalar b. This new generalisation of metric spaces is very interesting since octonions are not even a ring since they do not have the associative property of multiplication and the spaces do not satisfy the standard triangle inequality. Through these concepts, statistical convergence and related concepts are generalised. Properties associated with these concepts are given and the connections between them are established. Moreover, the influence of some structures of octonions on statistical convergence is analysed. Octonion-valued structures extend beyond quaternion-valued frameworks by incorporating non-associative properties, offering a richer setting for studying convergence phenomena. The present work provides new insights by establishing results that are not obtainable in associative settings. These and similar facts make the results obtained in these defined b-metric spaces of particular interest.